Different variables, one observation each

4 A certain make of washing machine has a wash-time with mean 56.9 minutes and standard deviation 4.8 minutes. A certain make of tumble dryer has a drying-time with mean 61.1 minutes and standard deviation 6.3 minutes. Both times are normally distributed and are independent of each other. Find the probability that a randomly chosen wash-time differs by more than 3 minutes from a randomly chosen drying-time.

5 The marks in paper 1 and paper 2 of an examination are denoted by \(X\) and \(Y\) respectively, where \(X\) and \(Y\) have the independent continuous distributions \(\mathrm { N } \left( 56,6 ^ { 2 } \right)\) and \(\mathrm { N } \left( 43,5 ^ { 2 } \right)\) respectively.

1. Find the probability that a randomly chosen paper 1 mark is more than a randomly chosen paper 2 mark.
2. Each candidate's overall mark is \(M\) where \(M = X + 1.5 Y\). The minimum overall mark for grade A is 135 . Find the proportion of students who gain a grade A .

2 In a Year 8 internal examination in a large school the Geography marks, \(G\), and Mathematics marks, \(M\), had means and standard deviations as follows. Assuming that \(G\) and \(M\) have independent normal distributions, find the probability that a randomly chosen Geography candidate scores at least 10 marks more than a randomly chosen Mathematics candidate. Do not use a continuity correction.

6. A potter makes decorative tiles in two colours, red and yellow. The length, \(R \mathrm {~cm}\), of the red tiles has a normal distribution with mean 15 cm and standard deviation 1.5 cm . The length, \(Y \mathrm {~cm}\), of the yellow tiles has the normal distribution \(\mathrm { N } \left( 12,0.8 ^ { 2 } \right)\). The random variables \(R\) and \(Y\) are independent. A red tile and a yellow tile are chosen at random.

1. Find the probability that the yellow tile is longer than the red tile. Taruni buys 3 red tiles and 1 yellow tile.
2. Find the probability that the total length of the 3 red tiles is less than 4 times the length of the yellow tile. Stefan defines the random variable \(X = a R + b Y\), where \(a\) and \(b\) are constants. He wants to use values of \(a\) and \(b\) such that \(X\) has a mean of 780 and minimum variance.
3. Find the value of \(a\) and the value of \(b\) that Stefan should use. \includegraphics[max width=\textwidth, alt={}, center]{ba3f3f9c-53d2-4e95-b2f3-3f617f1821ed-19_2255_50_314_34}

1. A baker produces bread buns and bread rolls. The weights of buns, \(B\) grams, and the weights of rolls, \(R\) grams, are such that \(B \sim \mathrm {~N} \left( 55,1.3 ^ { 2 } \right)\) and \(R \sim \mathrm {~N} \left( 51,1.2 ^ { 2 } \right)\)

A bun and a roll are selected at random.

1. Find the probability that the bun weighs less than \(110 \%\) of the weight of the roll. Two buns are chosen at random.
2. Find the probability that their weights differ by more than 1 gram. The baker sells bread in bags. Each bag contains either 10 buns or 11 rolls. The weight of an empty bag, \(S\) grams, is such that \(S \sim \mathrm {~N} \left( 3,0.2 ^ { 2 } \right)\)
3. Find the probability that a bag of buns weighs less than a bag of rolls.

6 The weight, \(X\) grams, of a dressed pheasant may be modelled by a normal random variable with a mean of 1000 and a standard deviation of 120 . Pairs of dressed pheasants are selected for packing into boxes. The total weight of a pair, \(Y = X _ { 1 } + X _ { 2 }\) grams, may be modelled by a normal distribution with a mean of 2000 and a standard deviation of 140 .

1. 1. Show that \(\operatorname { Cov } \left( X _ { 1 } , X _ { 2 } \right) = - 4600\).
   2. Given that \(X _ { 1 } - X _ { 2 }\) may be assumed to be normally distributed, determine the probability that the difference between the weights of a selected pair of dressed pheasants exceeds 250 grams.
2. The weight of a box is independent of the total weight of a pair of dressed pheasants, and is normally distributed with a mean of 500 grams and a standard deviation of 40 grams. Determine the probability that a box containing a pair of dressed pheasants weighs less than 2750 grams.
   \includegraphics[max width=\textwidth, alt={}]{fa3bf9d6-f064-4214-acff-d8b88c33a81e-16_2486_1714_221_153}

5. A child is playing with a set of red and blue wooden cubes. The side length of the red cubes is normally distributed with a mean of 14.5 cm and a variance of \(16.0 \mathrm {~cm} ^ { 2 }\). The side length of the blue cubes is normally distributed with a mean of 12.2 cm and a variance of \(9.0 \mathrm {~cm} ^ { 2 }\).

1. Find the probability that a randomly chosen red cube will have a side length of more than 3 cm greater than a randomly chosen blue cube. The child makes two towers, one from 4 red cubes and one from 5 blue cubes. Assuming that the cubes for each colour of tower were chosen at random,
2. find the probability that the red tower is taller than the blue tower.
3. Explain why the assumption that the cubes for each tower were chosen at random is unlikely to be realistic.

4 It is given that \(X\) and \(Y\) are independent random variables with distributions \(\mathrm { N } \left( \mu _ { x } , \sigma _ { x } ^ { 2 } \right)\) and \(\mathrm { N } \left( \mu _ { y } , \sigma _ { y } ^ { 2 } \right)\) respectively, and that \(W\) is a random variable such that \(W = X + Y\).

1. Use moment generating functions to show that the distribution of \(W\) is \(\mathrm { N } \left( \mu _ { x } + \mu _ { y } , \sigma _ { x } ^ { 2 } + \sigma _ { y } ^ { 2 } \right)\).
2. State the distribution of \(X - Y\). The diameters of the central poles of one brand of rotary clothes lines are normally distributed with mean 3.75 cm and variance \(0.000125 \mathrm {~cm} ^ { 2 }\). The diameters of the cylindrical tubes, into which the central poles fit, are normally distributed with mean 3.85 cm and variance \(0.0001 \mathrm {~cm} ^ { 2 }\). Poles and tubes are chosen at random. The 'clearance' between a tube and a pole is the diameter of the tube minus the diameter of the pole.
3. Find the probability that a pole and tube have a clearance between 0.08 cm and 0.13 cm .
4. Given that a pole and tube have a clearance between 0.08 cm and 0.13 cm , find the probability that the clearance is between 0.11 cm and 0.125 cm .